IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 64, NO. 12, DECEMBER 2016 4341 New Single-Source Surface Integral Equation for Magneto-Quasi-Static Characterization of Transmission Lines Situated in Multilayered Media Shucheng Zheng, Student Member, IEEE, Anton Menshov, Student Member, IEEE, and Vladimir I. Okhmatovski, Senior Member, IEEE Abstract— We recently proposed a novel single-source integral equation (SSIE) for accurate broadband resistance and inductance extraction and current flow modeling in 2-D conductors. The new surface integral equation is advanta- geous compared with the traditional volume electric field integral equation (V-EFIE) used for the inductance extraction, since the unknown function is defined on the surface of conductors as opposed to the volumetric unknown current density in V-EFIE. The new SSIE is also more suitable for the solution of inductance extraction problems than the traditional surface integral equation formulations, as it features only a single unknown surface function as opposed to having the unknown equivalent electric and magnetic surface current densities. The new equation also features only the electric field Green’s functions unlike the previously known SSIE formulations. The latter property makes the new SSIE equation particularly suitable to the inclusion of the multilayered substrate effect into the inductance extraction model. This paper describes the generalization of the new SSIE formulation to the case of transmission line models embedded into the multilayered lossy substrates. This paper also shows how the matrix sparsity in the method of moments discretization of the novel integral equation can be exploited to accelerate its numerical solution and reduce associated memory use. This sparsity arises due to the skin-effect-based attenuation of the fields in conductors’ cross sections leading to vanishing levels of the matrix elements corresponding to the distant interactions. Typical examples of inductance extraction in complex intercon- nects situated in lossy substrate are considered to validate the proposed techniques against traditional approaches. Index Terms— Inductance extraction, multiconductor transmission lines (MTLs), multilayered media, single-source integral equations (SSIEs). Manuscript received July 2, 2016; revised October 11, 2016 and October 26, 2016; accepted October 27, 2016. Date of publication November 21, 2016; date of current version December 7, 2016. This work was supported by a Collaborative Research and Development Grant from the Natural Sciences and Engineering Research Council (NSERC) and by the Manitoba HVDC Research Center of Manitoba Hydro International. An earlier version of this paper was presented at the IEEE MTT-S International Microwave Symposium, San Francisco, CA, USA, May 22–27, 2016. S. Zheng and V. I. Okhmatovski are with the Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, MB R3T 5V6, Canada (e-mail: umzheng6@myumanitoba.ca; vladimir.okhmatovski@umanitoba.ca). A. Menshov is with the Department of Electrical and Computer Engineer- ing, The University of Texas at Austin, Austin, TX 78712 USA (e-mail: anton.menshov@utexas.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2016.2623625 I. I NTRODUCTION T RANSIENT analysis of multiconductor transmission lines (MTLs) plays an important role in the design of high-speed digital interconnects [1]–[3], analysis of microwave and millimeter wave circuits [4], simulation of power systems [5]–[7], and various other areas. Under the assump- tion of quasi-TEM wave propagation along an MTL situated in lossy dielectric substrate, the Maxwell equations are simplified to the system of Telegrapher’s equations governing the wave propagation along the line, and the decoupled problems of electro- and magneto-quasi-static governing cross-sectional components of the electric and magnetic fields, respectively [4], [8], [9]. Solution of the electro- and magneto-quasi-static problems yields frequency-dependent per-unit-length (p.u.l.) capacitance (C ), conductance (G), inductance ( L ), and resistance ( R) matrices, which upon substitution into the Telegrapher’s equations enable tran- sient analysis of signal propagation along the MTL conductors. This paper presents a new surface integral equation formulation for magneto-quasi-static analysis of the MTL embedded into lossy dielectric substrates. Such analysis is traditionally done via solution of the volume electric field integral equation (V-EFIE) under magneto-quasi-static approx- imation [3], [10]. In our previous works [11]–[13], we pro- posed novel surface-volume-surface EFIEs (SVS-EFIE) for the broadband network characterization and current flow modeling in 2-D conductors and 3-D interconnects of arbitrary cross section. The novel equations are derived from the classical V-EFIE [14] through the representation of the electric field inside a conductor as a superposition of the cylindrical waves emanating from the conductor’s surface. Thus, derived single- source integral equations (SSIEs) feature only the derivative- free electric field kernel and confine the unknowns in the method of moments (MoM) discretization to the contour of the conductor. The latter property greatly reduces the compu- tational complexity of the numerical solution compared with the solution of the traditional V-EFIE. Alternative single-source surface integral equa- tions [15], [16] feature similar number of unknowns as the proposed SVS-EFIE, though they introduce large number of 0018-9480 © 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.